In short, dragon-curves are organic and lively, quite unlike the rigid, lifeless solidity of a square. But there’s more to a dragon-curve than immediately meets the eye. Dragon-curves are rep-tiles, that is, you can tile one with smaller copies of itself:

Dragon-curve rep-tiled with two copies of itself

Dragon-curve rep-4

Dragon-curve rep-8

Dragon-curve rep-16

Dragon-curve rep-32

Dragon-curve self-tiling (animated)

From the rep-32 dragon-curve, you can see that a dragon-curve can be surrounded by six copies of itself. Here’s an animation of the process:

Dragon-curve surrounded (anim)

And because dragon-curves are rep-tiles, they will tile the plane:

Dragon-curve tiling #1

Dragon-curve tiling #2

But how do you make these strange and beautiful shapes, with their myriad curves and curlicules, their energy and liveliness? It’s actually very simple. You start with the shape generally regarded as the dullest and most everyday of all:

A square

Then you see how the shape can be replaced by five smaller copies of itself: